constructoragdatheorem-provingdisjoint-union

Are constructors disjoint in Agda? (or how to disprove inj₁ x ≡ inj₂ y)

I need one more lemma showing that inj₁ x ≡ inj₂ y is absurd as part of a larger theorem about disjoint union types (⊎) in Agda.

This result would follow directly from the two constructors for ⊎, namely inj₁ and inj₂, being disjoint. Is that the case in Agda? How do I prove it?

Here is the complete lemma:

open import Relation.Nullary
open import Relation.Binary.PropositionalEquality
open import Data.Sum


lemma : ∀ {a b} {A : Set a} {B : Set b} {x : A} {y : B} → ¬ inj₁ x ≡ inj₂ y
lemma eq = ?
Solution

  • The data type constructors are disjoint. I'd say it's a theorem in Agda's type-system meta-theory.

    You can try to case the eq proof (C-c C-c), and Agda will find the contradiction:

    lemma : ∀ {a b} {A : Set a} {B : Set b} {x : A} {y : B} → ¬ inj₁ x ≡ inj₂ y
lemma ()

    This happily type-checks.